Skip to main content
SpringerLink
Log in

Difference Forms

  • Published:
Foundations of Computational Mathematics Aims and scope Submit manuscript

Abstract

Currently, there is much interest in the development of geometric integrators, which retain analogues of geometric properties of an approximated system. This paper provides a means of ensuring that finite difference schemes accurately mirror global properties of approximated systems. To this end, we introduce a cohomology theory for lattice varieties, on which finite difference schemes and other difference equations are defined. We do not assume that there is any continuous space, or that a scheme or difference equation has a continuum limit. This distinguishes our approach from theories of “discrete differential forms” built on simplicial approximations and Whitney forms, and from cohomology theories built on cubical complexes. Indeed, whereas cochains on cubical complexes can be mapped injectively to our difference forms, a bijection may not exist. Thus our approach generalizes what can be achieved with cubical cohomology. The fundamental property that we use to prove our results is the natural ordering on the integers. We show that our cohomology can be calculated from a good cover, just as de Rham cohomology can. We postulate that the dimension of solution space of a globally defined linear recurrence relation equals the analogue of the Euler characteristic for the lattice variety. Most of our exposition deals with forward differences, but little modification is needed to treat other finite difference schemes, including Gauss-Legendre and Preissmann schemes.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. D. Arnold, Differential complexes and numerical stability, Plenary address at ICM Beijing, 2002. http://www.ima.umn.edu/~arnold/papers/icm2002.pdf.

  2. D. Arnold, R. S. Falk, and R. Winthur, Finite element exterior calculus, homological techniques and applications, Acta Numerica 15 (2006), 1–155.

    Article  MATH  MathSciNet  Google Scholar 

  3. A. Bossavit, Differential forms and the computation of fields and forces in electromagnetism, Eur. J. Mech. B-Fluids 10 (1991), 474–488.

    MATH  MathSciNet  Google Scholar 

  4. A. Bossavit, Whitney forms—a class of finite-elements for 3-dimensional computations in electromagnetism, IEE Proc. A 135 (1988), 493–500.

    Google Scholar 

  5. R. Bott and L. W. Tu, Differential Forms in Algebraic Topology, Graduate Texts in Mathematics, Vol. 82, Springer-Verlag, New York, 1982.

    MATH  Google Scholar 

  6. T. J. Bridges and S. Reich, Multi-symplectic integrators: numerical schemes for Hamiltonian PDEs that preserve symplecticity, Phys. Lett. A 284 (2001), 184–193.

    Article  MATH  MathSciNet  Google Scholar 

  7. C. J. Budd and G. J. Collins, An invariant moving mesh scheme for the nonlinear diffusion equation, App. Num. Math. 26 (1998), 23–29.

    Article  MATH  MathSciNet  Google Scholar 

  8. C. J. Budd and M. D. Piggott, Geometric integration and its applications, in Handbook of Numerical Analysis, Vol. XI, pp. 35–139, North-Holland, Amsterdam, 2003.

    Google Scholar 

  9. J. A. Chard and V. Shapiro, A multivector data structure for differential forms and equations, Math. Comp. Sim. 54 (2000), 33–64.

    Article  MathSciNet  Google Scholar 

  10. R. Forman, Morse theory for cell complexes, Topology 37 (1998), 945–979.

    Article  MATH  MathSciNet  Google Scholar 

  11. T. Frankel, The Geometry of Physics, Cambridge University Press, Cambridge, 1997.

    MATH  Google Scholar 

  12. W. Fulton, Algebraic Topology, A First Course, Graduate Texts in Mathematics, Vol. 153, Springer-Verlag, New York, 1995.

    MATH  Google Scholar 

  13. G. Zhong and J. E. Marsden, Lie-Poisson Hamilton-Jacobi theory and Lie-Poisson integrators, Phys. Lett. A 133 (1988), 134–139.

    Article  MathSciNet  Google Scholar 

  14. P. W. Gross and P. R. Kotiuga, Electromagnetic Theory and Computation: A Topological Approach, MSRI Publications, Vol. 48, Cambridge University Press, Cambridge, 2004.

    MATH  Google Scholar 

  15. G. Chaohao (ed.), Soliton Theory and its Applications, Springer-Verlag, Berlin, 1995.

    MATH  Google Scholar 

  16. E. Hairer, C. Lubich, and G. Wanner, Geometric Numerical Integration, Springer Series in Computational Mathematics, Vol. 31, Springer-Verlag, Berlin, 2002.

    MATH  Google Scholar 

  17. R. Hiptmair, Finite elements in computational electromagnetism, Acta Numerica 11 (2002), 237–340.

    Article  MATH  MathSciNet  Google Scholar 

  18. P. E. Hydon, Symmetries and first integrals of ordinary difference equations, Proc. Roy. Soc. A 456 (2000), 2835–2855.

    Article  MATH  MathSciNet  Google Scholar 

  19. P. E. Hydon, Conservation laws of partial difference equations with two independent variables, J. Phys. A: Math. Gen. 34 (2001), 10347–10355.

    Article  MATH  MathSciNet  Google Scholar 

  20. P. E. Hydon and E. L. Mansfield, A variational complex for difference equations, J. Found. Comp. Math. 4 (2004), 187–217.

    Article  MATH  MathSciNet  Google Scholar 

  21. A. Iserles, H. Munthe-Kaas, S. Norsett, and A. Zanna, Lie group methods, Acta Numerica 9 (2000), 215–365.

    Article  MathSciNet  Google Scholar 

  22. T. Kaczynski, K. Mischaikow, and M. Mrozek, Computing homology, Homology, Homotopy and Applications 5 (2003), 233–256.

    MATH  MathSciNet  Google Scholar 

  23. L. Kharevych, W. Y. Tong, E. Kanso, J. E. Marsden, P. Schröder, and M. Desbrun Geometric, variational integrators for computer animation, in Eurographics/ACM SIGGRAPH Symposium on Computer Animation (M.-P. Cani and J. O’Brien, eds.), pp. 43–51, ACM, Portland, 2006.

    Google Scholar 

  24. T. Y. Kong, R. D. Kopperman, and P. R. Meyer, A topological approach to digital topology, Amer. Math. Monthly 98 (1991), 901–917.

    Article  MATH  MathSciNet  Google Scholar 

  25. M. Leok, Foundations of computational geometric mechanics, PhD Thesis, California Institute of Technology, 2004.

    Google Scholar 

  26. R. I. McLachlan and G. R. W. Quispel, Six lectures on the geometric integration of ODEs, in Foundations of Computational Mathematics (R. A. DeVore, A. Iserles, and E. Süli, eds.), London Mathematical Society Lecture Note Series, Vol. 284, pp. 155–210, Cambridge University Press, Cambridge, 2001.

    Google Scholar 

  27. E. L. Mansfield and P. E. Hydon, Towards approximations which preserve integrals, in Proc. ISSAC 2001 (B. Mourrain, ed.), pp. 217–222, ACM Publishing, New York, 2001.

    Chapter  Google Scholar 

  28. W. S. Massey, Singular Homology Theory, Graduate Texts in Mathematics, Vol. 70, Springer-Verlag, Berlin, 1980.

    MATH  Google Scholar 

  29. C. Mattiussi, An analysis of finite volume, finite element, and finite difference methods using some concepts from algebraic topology, J. Comp. Phys. 133 (1997), 289–309.

    Article  MATH  MathSciNet  Google Scholar 

  30. A. R. Mohebalhojeh, On shallow water potential vorticity inversion by Rossby-number expansions, Q. J. Roy. Met. Soc. 128 (2002), 679–694.

    Article  Google Scholar 

  31. J. M. Sanz-Serna and M. P. Calvo, Numerical Hamiltonian Problems, Chapman and Hall, London, 1994.

    MATH  Google Scholar 

  32. W. Schwalm, B. Moritz, M. Giona, and M. Schwalm, Vector difference calculus for physical lattice models, Phys. Rev. E 59 (1999), 1217–1233.

    Article  MathSciNet  Google Scholar 

  33. S. Suuriniemi and L. Kettunen, Trade-off between information and complexity: a technique for automated topological computations, COMPEL 22 (2003), 481–494.

    MATH  MathSciNet  Google Scholar 

  34. Y. Y. Tong, S. Lombeya, A. N. Hirani, and M. Desbrun, Discrete multiscale vector field decomposition, ACM Trans. Graphics 22 (2003), 445–452.

    Article  Google Scholar 

  35. E. Tonti, On the formal structure of physical theories, Istituto di Matematica del Politecnico di Milano, Milan, http://www.dic.univ.trieste.it/perspage/tonti/papershtm, 1975.

  36. A. Weil, Sur les théorèmes de De Rham, Comment. Math. Hel. 26 (1952), 119–145.

    Article  MATH  MathSciNet  Google Scholar 

  37. H. Whitney, Geometric Integration Theory, Princeton University Press, Princeton, 1957.

    MATH  Google Scholar 

  38. D. K. Wise, p-form electromagnetism on discrete space-times, Class. Quantum Grav. 23 (2006), 5129–5176.

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institute of Mathematics, Statistics and Actuarial Science, University of Kent, Canterbury, CT2 7NF, UK

    Elizabeth L. Mansfield

  2. Department of Mathematics and Statistics, University of Surrey, Guildford, GU2 7XH, UK

    Peter E. Hydon

Authors
  1. Elizabeth L. Mansfield
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Peter E. Hydon
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Elizabeth L. Mansfield.

Additional information

Dedicated to Professor Arieh Iserles on the Occasion of his Sixtieth Birthday.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Mansfield, E.L., Hydon, P.E. Difference Forms. Found Comput Math 8, 427–467 (2008). https://doi.org/10.1007/s10208-007-9015-8

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10208-007-9015-8

Keywords

Mathematics Subject Classification (2000)

Search

Navigation